A certain escalator travels
at a rate of 12.1 meters per minute,
and its angle of inclination is 27.1 degrees.
We're asked to determine the vertical component
of the velocity and then how long it will take a passenger
to travel 13.1 meters vertically.
Let's represent the given information using a vector
where because the angle of inclination is 27.1 degrees,
the angle between the positive x-axis and the vector
is 27.1 degrees.
Let's say the vector is this vector here,
which means this angle here is 27.1 degrees.
This is called the direction of the vector.
We have to use the angle theta to represent this angle.
And because the escalator is traveling
at a rate of 12.1 meters per minute,
the magnitude or length of the vector is 12.1.
Let's also label that.
And now let's sketch the horizontal and vertical components
of this vector,
where this would be the horizontal component.
Let's label this x.
And this would be the vertical component.
Let's label this y.
To answer our question,
we need to find the vertical component,
but we'll go ahead and find both components
and explain the meaning of both.
Notice how by sketching the horizontal
and vertical components, we now have a right triangle
that we can use to solve for x and y.
Notice, the x-component is adjacent to the angle,
and we know the length of the hypotenuse is 12.1.
And therefore, we know that the cosine of 27.1 degrees
must equal x divided by 12.1.
We can solve for x by multiplying both sides by 12.1,
which gives us x equals 12.1
times cosine of 27.1 degrees.
Similarly, we can state that the sine of 27.1 degrees
is equal to the ratio of the opposite side
to the hypotenuse, which is y divided by 12.1.
Solving for y, we have y equals 12.1
times sine of 27.1 degrees.
And now by finding x and y,
we can determine the x- and y-components
of the given vector.
We could've found this a little bit faster
by using the equation shown below,
where if a vector v makes an angle theta
with the positive x-axis, then the x-component is equal to
the magnitude of v times cosine theta,
which is what we have here,
and the y-component is equal to
the magnitude of v times sine theta,
which is what we have here.
And now we'll go to the calculator.
And we are told to round to the 10ths place value.
We first want to make sure we are in Degree mode,
which we are.
And then we enter 12.1
cosine 27.1 degrees
and 12.1
sine 27.1 degrees.
Rounding to the 10ths,
the x-component is approximately 12.8,
and the y-component is approximately 5.5.
Now that we have the approximate components,
let's discuss what this means.
The horizontal or x-component of 10.8
indicates the escalator is traveling
at a rate of 10.8 meters per minute horizontally,
and the vertical or y-component of 5.5
indicates the escalator is traveling vertically or up
at a rate of 5.5 meters per minute.
So for the first question,
the vertical component of the velocity is 5.5
meters per minute.
The second question is how long will it take a passenger
to travel 13.1 meters vertically.
For this, we need to remember the formula distance
equals rate times time.
The vertical distance is 13.1 meters,
and the vertical rate is the rate of 5.5 meters per minute.
So we have 5.5 for r, and t is time in minutes.
To solve for t, we divide both sides by 5.5.
Simplifying,
t is approximately 2.4,
which means it will take a passenger
approximately 2.4 minutes
to travel 13.1 meters vertically.
I hope you found this helpful.